I started class by assigning each group 2 of the 4 problems (everyone did #4). I gave them 10 minutes to brainstorm on how to approach the problem. The first group who got their equation explored the problem by using a table and finding the rate of change. The two points given in this problem were (0, 36) and (6, 24) so they remembered that the starting point had to be 36 because we had discussed for several days that the x-coordinate of the starting point is always 0. They realized the values for y went down by 2 each time and wrote the equation. One of the girls actually said, "I am getting good at this!" Here are their In-Out tables:
After the first 10 minutes I had the girls share with the class how they came to find their equation for #2. I took a graph from one of the students in a group that was working on #3 where the 2 points given were (2, 300) and (10, 196). He had graphed the 2 points and drawn a line through it which touched both the x and y axes. By looking at the graph the starting point was estimated to be 325. I gave the groups another 10 minutes to work. There was another group working on #3 and they were trying to build a table that worked with a starting point of 325. One of the students at that group blurted out, "I got the rate of change. It's 13." I directed the students to try that rate of change and adjust the starting point as needed and they found an equation that worked! I was surprised that the student's rate of change was correct. I went to the boy's side and asked how he found it. He had subtracted the y values and then divided the answer by the difference of the x values!!! Shazam! He used the slope formula and didn't even know it! The slope was actually -13 instead of 13 but he figured that out when he wrote his equation because the y values were getting smaller. I had him share with the class how he found the slope and then I went to the board and wrote the slope formula there. I asked them if they knew what it was and none of them remembered it from last year. I showed them how my students from previous years found the slope using the formula. I also showed them that if they "plug into" the formula correctly they would get the correct sign for the slope of the line. I was so excited at the connections that we were making to the "traditional" algebra. The majority of the groups came up with the correct equations...on the first day that we wrote equations given 2 points!!! I was amazed!
This is Hunter sharing how he found the rate of change using 2 points. He did an incredible job of figuring it out AND explaining it!
Lastly, #4 in this activity was the first situation/problem that we have encountered where the starting point is negative. The 2 points given were (3, 12) and (7,32). I had 2 different students to use a table to find that the rate of change is positive 5. I had to give them a prompt to help them to find the starting point. The students who figured out the rate of change had created tables that started with an input value of 3 and went up to 7. All I had to do was put a 2, 1, and 0 in the input column (I put them above the 3 to help them see the pattern) and asked them if they could continue the pattern to find the starting point. It ended up being -3.
Now, my teaching buddy, Mrs. New, and I have been talking about whether or not teaching the formulas is important. AND...we still think that students need to be exposed to the "old-school" formulas and even practice some problems using the formulas. However...I think I had more students to get their equations correct given 2 points (on the 1st day!!!) than I ever have. The exploration and work that we have been doing over the last several weeks gives them such a firm conceptual understanding of writing equations by finding the starting point and rate of change that I believe the "old-school" work will have more meaning!!! Mrs. New keeps asking me to consider whether or not the majority of our students remember how to use the formulas when they see these problems on standardized tests and my answer is NO... the majority do not remember! Now our students have a "hook" (as our instructional partner, Dr. Montgomery calls it) on which to hang that concept so that they will hopefully retain the information. They have also been shown how to use various approaches to examine and solve these problems.